Measurement Units as Standard

In the history of measurement we see a great deal of attention given to the matter of standardizing the measurement units. The English Customary system of measurement has a long and colorful history of determining, defining, and re-defining measurement units. Originally the various units emerged from one tradition or another, and eventually the power and authority of a government were applied in order to eliminate variations on the sizes of the units. For example, the mile is said to have originated in ancient Rome where it was defined as "the length of 1,000 paces of a Roman legion" (Rowlett, 2001). Since the "pace" in question was about 5 feet (the right step and the left step together), the mile was about 5,000 feet long. For most of human history there was no need for more precision than this because transportation over long distances took such a long time. There was no practical significance to the difference between 50 miles and 51 miles: both took about two or three days to walk! By sixteenth-century England there were apparently a variety of slightly different mile-lengths in use, so in 1592 Parliament decreed that the official length of the mile would be "8 furlongs." Because a British furlong was 660 feet, the length of the mile turned out to be 5,280 feet.

Vocabulary Lesson

A smidgen is a very small quantity of material. Until recently, no one though a smidgen was an actual unit of measure, but recently kitchen supply stores in the U.S. and other countries have begun selling sets of "minispoons" in which the smallest spoon, labeled "smidgen," is designed to hold exactly 1⁄2 pinch or 1⁄32 teaspoon, which is roughly 0.005 fluid ounce or 0.15 milliliter. The word is a diminutive of "smutch" or "smudge"; it originally meant a small spot.

The issue of definition was not complete, however. Acts such as this one by Parliament could legally define an equivalence (1 mile = 8 furlongs), but this is a relative standard rather than an absolute standard. In other words, we know how many furlongs to a mile, but how long, exactly, is a furlong? Well, another equivalence could be cited (1 furlong = 660 feet), but that is still a relative definition. How long is a foot? This chasing of absolute standards began to come to an end with the invention of the metric system. In the late eighteenth century, the French realized that some form of absolute measurement unit was needed. They invented a basic unit of measurement of length, called the meter, which would be one ten-millionth of the distance between the North Pole and the Equator. Now, although this is still not an absolute standard, it is the closest anyone has ever come to establishing one. What could be more absolute than the size of the earth itself? The French built an entire system based on composing and decomposing units with powers of ten. Units of volume were also determined in relation to the meter. (A liter was to be the volume contained in a cube that was meter on each side.) Units of weight were also based on this same standard. (A kilogram was to be the weight of a liter of water.) Since the development of the metric system, governments around the world have either adopted the metric system or redefined their customary units of measurement in reference to metric units. This effort has led to the International System of Units,3 which is a network of international treaties and legal definitions of measurement units in relation to metric units. For example, today's official definition of a mile is 1,609.344 meters. Since this is tied to the standard of the meter, which is defined in reference to the size of the earth, we have the closest thing to an absolute standard that humans are likely to construct.

Elementary Mathematics: Pedagogical Content Knowledge, by James E. Schwartz, is designed to sharpen pre-service and in-service teachers' mathematics pedagogical content knowledge. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the text thoroughly integrates discussion of the five NCTM process strands.

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